arXiv research

Compactifies maximal component of surface group representations into a closed ball.

Diameters of ball intersections decrease as centers move apart.

We obtain asymptotics of sequences of the holomorphic sections of the pluricanonical bundles on ball quotients associated to closed geodesics. A nonvanishing result follows.

Adding a point to configurations in closed balls depends on the number of points and their ordering.

In a compact orbifold, for small prescribed volume, an isoperimetric region is close to a small metric ball; in a Euclidean orbifold, it is a small metric ball.

Compactness theorem for manifolds with scalar curvature and entropy bounds.

Symplectic capacities of domains near balls are well-defined, but not for all $C^1$-close domains.

The study explores conditions for a manifold to be a leaf of a complete foliation on the open unit ball.

Small sub-Riemannian balls have diameter close to twice their radius.

Stable capillary surfaces in weighted balls are disks.

4-ball can be tiled with knotted surfaces.

In this note we provide natural optimal geometric conditions for a Riemannian manifold suitably covered by two open metric balls to be homeomorphic to a sphere. This can be viewed as a geometric analogue of Brown's theorem in topology stating that a closed manifold covered by two topological balls is a sphere.

Spaces of polynomials are shown to be Euclidean balls.

We obtain an improved pseudolocality result for Ricci flows on two-dimensional surfaces that are initially almost-hyperbolic on large hyperbolic balls. We prove that, at the central point of the hyperbolic ball, the Gauss curvature remains close to the hyperbolic value for a time that grows exponentially in the radius …

Article provides polytopes as dual unit balls of Thurston norms on 3-manifolds.

We prove that three spaces of importance in topological combinatorics are homeomorphic to closed balls: the totally nonnegative Grassmannian, the compactification of the space of electrical networks, and the cyclically symmetric amplituhedron.

New theorem shows shapes close to balls, flow converges to balls in 2D and 3D.

This paper studies certain embedded spheres in closed affine manifolds. For $n \geq 3$, we investigate the dome bodies in a closed affine $n$-manifold $M$ with its boundary homeomorphic to a sphere under the assumption that a developing map restricted to a component of $\partial\hat{M}$ is an embedding onto a strictly …

We show that under reasonable conditions, the spines of the handlebodies of a strongly irreducible Heegaard splitting will intersect a closed ball in a graph which is isotopic into the boundary of the ball. This is in some sense a generalization of the results by Scharlemann on how a strongly irreducible Heegaard split…

The paper proves the stability of a 3-ball under curvature constraints.

We show that, the solutions of the isoperimetric problem for small volumes are $C^{2,α}$-close to small spheres. On the way, we define a class of submanifolds called pseudo balls, defined by an equation weaker than constancy of mean curvature. We show that in a neighborhood of each point of a compact riemannian manifol…

Growth rate of Dehn twist lattice points in Teichmüller space is slower than mapping class group lattice points.

Survey of rigidity and gap phenomena in sphere-ball submanifolds.

New symplectic caps and embeddings found in complex projective plane.

Constructs minimal surfaces near the boundary of a ball.

We construct non-trivial continuous isospectral deformations of Riemannian metrics on the ball and on the sphere in $\R^n$ for every $n\geq 9$. The metrics on the sphere can be chosen arbitrarily close to the round metric; in particular, they can be chosen to be positively curved. The metrics on the ball are both Diric…

Study minimal networks on spheres and balls near standard metrics.

The minimal number of critical points is studied for smooth functions on closed manifolds.

If $(M^n, g)$ is a closed Riemannian manifold where every unit ball has volume at most $ε_n$ (a sufficiently small constant), then the $(n-1)$-dimensional Uryson width of $(M^n, g)$ is at most 1.

If (M^n, g) is a complete Riemannian manifold with filling radius at least R, then we prove that it contains a ball of radius R and volume at least c(n)R^n. If (M^n, hyp) is a closed hyperbolic manifold and if g is another metric on M with volume at most c(n)Volume(M,hyp), then we prove that the universal cover of (M,g…

The paper divides minimal hypersurfaces in a ball into two parts.

Smooth 4-manifolds have simple horizontal decompositions.

The paper divides minimal surfaces in a ball into two parts.

Holomorphic foliations found in ball space with unique properties.

In this paper we prove that the unit ball $\mathbb{B}$ of $\mathbb{C}^2$ admits complete properly embedded complex curves of any given topological type. Moreover, we provide examples containing any given closed discrete subset of $\mathbb{B}$.

Localized sum-of-norms clustering separates balls in data.

We discuss closed symplectic 4-manifolds which admit full symplectic packings by $N$ equal balls for large $N$'s. We give a homological criterion for recognizing such manifolds. As a corollary we prove that ${\Bbb C}P^2$ can be fully packed by $N$ equal balls for every $N\geq 9$.

We show if M is a closed, connected, orientable, hyperbolic 3-manifold with Heegaard genus g then g >= 1/2 cosh(r) where r denotes the radius of any isometrically embedded ball in M. Assuming an unpublished result of Pitts and Rubinstein improves this to g >= 1/2 cosh(r) + 1/2. We also give an upper bound on the volume…

Study shows Seifert fibered spaces don't bound rational homology balls.

We prove the existence of extremal domains for the first eigenvalue of the Laplace-Beltrami operator in some compact Riemannian manifolds of dimension $n \geq 2$, with volume close to the volume of the manifold. If the first (positive) eigenfunction $φ_0$ of the Laplace-Beltrami operator over the manifold is a nonconst…

In this paper, we prove uniform lower bounds on the volume growth of balls in the universal covers of Riemannian surfaces and graphs. More precisely, there exists a constant $δ>0$ such that if $(M,hyp)$ is a closed hyperbolic surface and $h$ another metric on $M$ with $\area(M,h)\leq δ\area(M,hyp)$ then for every radiu…

Study geodesic structure of compact balls space and find explicit isometry.

The study finds at least 2 free-boundary minimal disks in convex 3-balls.

The paper proves and analyzes Minkowski inequalities for nearly spherical domains.

We construct pairs of conformally equivalent isospectral Riemannian metrics $φ_1 g$ and $φ_2 g$ on spheres $S^n$ and balls $B^{n+1}$ for certain dimensions $n$, the smallest of which is $n=7$, and on certain compact simple Lie groups. In the case of Lie groups, the metric $g$ is left-invariant. In the case of spheres a…

Study pinches eigenvalues and curvatures of convex hypersurfaces to make them nearly geodesic spheres.

Dancing polygons and rolling balls linked via a special geometric distribution.

Given a closed subset $\La$ of the open unit ball $B_1\subset \real^n$, $n \geq 3$, we will consider a complete Riemannian metric $g$ on $\bar{B_1} \setminus \La$ of constant scalar curvature equal to $n(n-1)$ and conformally related to the Euclidean metric. In this paper we prove that every closed Euclidean ball $\bar…